Daftar integral dari fungsi rasional


Dari Wikipedia Indonesia, artikel bebas.

 

int (ax + b)^n dx  = frac{(ax + b)^{n+1}}{a(n + 1)} qquadmbox{(untuk } nneq -1mbox{)},!
intfrac{1}{ax + b} dx  = frac{1}{a}lnleft|ax + bright|
int x(ax + b)^n dx  = frac{a(n + 1)x - b}{a^2(n + 1)(n + 2)} (ax + b)^{n+1} qquadmbox{(untuk }n notin {-1, -2}mbox{)}

 

intfrac{x}{ax + b} dx  = frac{x}{a} - frac{b}{a^2}lnleft|ax + bright|
intfrac{x}{(ax + b)^2} dx  = frac{b}{a^2(ax + b)} + frac{1}{a^2}lnleft|ax + bright|
intfrac{x}{(ax + b)^n} dx  = frac{a(1 - n)x - b}{a^2(n - 1)(n - 2)(ax + b)^{n-1}} qquadmbox{(untuk } nnotin {1, 2}mbox{)}

 

intfrac{x^2}{ax + b} dx  = frac{1}{a^3}left(frac{(ax + b)^2}{2} - 2b(ax + b) + b^2lnleft|ax + bright|right)
intfrac{x^2}{(ax + b)^2} dx  = frac{1}{a^3}left(ax + b - 2blnleft|ax + bright| - frac{b^2}{ax + b}right)
intfrac{x^2}{(ax + b)^3} dx  = frac{1}{a^3}left(lnleft|ax + bright| + frac{2b}{ax + b} - frac{b^2}{2(ax + b)^2}right)
intfrac{x^2}{(ax + b)^n} dx  = frac{1}{a^3}left(-frac{(ax + b)^{3-n}}{(n-3)} + frac{2b (a + b)^{2-n}}{(n-2)} - frac{b^2 (ax + b)^{1-n}}{(n - 1)}right) qquadmbox{(untuk } nnotin {1, 2, 3}mbox{)}

 

intfrac{1}{x(ax + b)} dx  = -frac{1}{b}lnleft|frac{ax+b}{x}right|
intfrac{1}{x^2(ax+b)} dx  = -frac{1}{bx} + frac{a}{b^2}lnleft|frac{ax+b}{x}right|
intfrac{1}{x^2(ax+b)^2} dx  = -aleft(frac{1}{b^2(ax+b)} + frac{1}{ab^2x} - frac{2}{b^3}lnleft|frac{ax+b}{x}right|right)
intfrac{1}{x^2+a^2} dx  = frac{1}{a}arctanfrac{x}{a},!
intfrac{1}{x^2-a^2} dx =
  • |a|mbox{)},!” />

 

intfrac{1}{ax^2+bx+c} dx =
  • 0mbox{)}” />
  •  -frac{2}{2ax+b}qquadmbox{(untuk }4ac-b^2=0mbox{)}
intfrac{x}{ax^2+bx+c} dx  = frac{1}{2a}lnleft|ax^2+bx+cright|-frac{b}{2a}intfrac{dx}{ax^2+bx+c}

 

intfrac{mx+n}{ax^2+bx+c} dx =
  • 0mbox{)}” />
  •  frac{m}{2a}lnleft|ax^2+bx+cright|-frac{2an-bm}{a(2ax+b)} ,,,,,,,,,, qquadmbox{(untuk }4ac-b^2=0mbox{)}

 

intfrac{1}{(ax^2+bx+c)^n} dx= frac{2ax+b}{(n-1)(4ac-b^2)(ax^2+bx+c)^{n-1}}+frac{(2n-3)2a}{(n-1)(4ac-b^2)}intfrac{1}{(ax^2+bx+c)^{n-1}} dx,!
intfrac{x}{(ax^2+bx+c)^n} dx= frac{bx+2c}{(n-1)(4ac-b^2)(ax^2+bx+c)^{n-1}}-frac{b(2n-3)}{(n-1)(4ac-b^2)}intfrac{1}{(ax^2+bx+c)^{n-1}} dx,!
intfrac{1}{x(ax^2+bx+c)} dx= frac{1}{2c}lnleft|frac{x^2}{ax^2+bx+c}right|-frac{b}{2c}intfrac{1}{ax^2+bx+c} dx

Fungsi rasional apapun dapat diintegrasikan melalui persamaan-persamaan diatas dengan memanfaatkan integrasi parsial, dengan menguraikan fungsi rasional menjadi penjumlahan fungsi-fungsi dalam bentuk

frac{ex + f}{left(ax^2+bx+cright)^n}.
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